#!/usr/bin/env python
# coding: utf-8

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from sympy import *
init_printing(use_latex='mathjax')
x, y, z = symbols('x,y,z')
r, theta = symbols('r,theta', positive=True)


# ## Matrices
# 
# The SymPy `Matrix` object helps us with small problems in linear algebra.

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rot = Matrix([[r*cos(theta), -r*sin(theta)],
              [r*sin(theta),  r*cos(theta)]])
rot


# ### Standard methods

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rot.det()


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rot.inv()


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rot.singular_values()


# ### Exercise
# 
# Find the inverse of the following Matrix:
# 
# $$ \left[\begin{matrix}1 & x\\y & 1\end{matrix}\right] $$

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# Create a matrix and use the `.inv` method to find the inverse



# ### Operators
# 
# The standard SymPy operators work on matrices

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rot * 2


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rot * rot


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v = Matrix([[x], [y]])
v


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rot * v


# ### Exercise
# 
# In the last exercise you found the inverse of the following matrix

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M = Matrix([[1, x], [y, 1]])
M


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M.inv()


# Now verify that this is the true inverse by multiplying the matrix times its inverse.  Do you get the identity matrix back?

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# Multiply `M` by its inverse.  Do you get back the identity matrix?



# ### Exercise
# 
# What are the eigenvectors and eigenvalues of `M`?

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# Find the methods to compute eigenvectors and eigenvalues. Use these methods on `M`



# ### NumPy-like Item access

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rot[0, 0]


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rot[:, 0]


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rot[1, :]


# ### Mutation
# 
# We can change elements in the matrix.

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rot[0, 0] += 1
rot


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simplify(rot.det())


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rot.singular_values()


# ### Exercise
# 
# Play around with your matrix `M`, manipulating elements in a NumPy like way.  Then try the various methods that we've talked about (or others).  See what sort of answers you get.

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# Play with matrices